Grothendieck Group Decategorifications and Derived Abelian Categories
The Grothendieck group is an interesting invariant of an exact category. It induces a decategorication from the category of essentially small exact categories (whose morphisms are exact functors) to the category of abelian groups. Similarly, the triangulated Grothendieck group induces a decategorica...
Main Author: | |
---|---|
Other Authors: | |
Language: | en |
Published: |
Université d'Ottawa / University of Ottawa
2015
|
Subjects: | |
Online Access: | http://hdl.handle.net/10393/33000 http://dx.doi.org/10.20381/ruor-2842 |
id |
ndltd-uottawa.ca-oai-ruor.uottawa.ca-10393-33000 |
---|---|
record_format |
oai_dc |
spelling |
ndltd-uottawa.ca-oai-ruor.uottawa.ca-10393-330002018-01-05T19:02:28Z Grothendieck Group Decategorifications and Derived Abelian Categories McBride, Aaron Savage, Alistair additive categories abelian categories exact categories triangulated categories Grothendieck groups category theory cochain complexes homotopy category cohomology bounded derived category derived functors The Grothendieck group is an interesting invariant of an exact category. It induces a decategorication from the category of essentially small exact categories (whose morphisms are exact functors) to the category of abelian groups. Similarly, the triangulated Grothendieck group induces a decategorication from the category of essentially small triangulated categories (whose morphisms are triangulated functors) to the category of abelian groups. In the case of an essentially small abelian category, its Grothendieck group and the triangulated Grothendieck group of its bounded derived category are isomorphic as groups via a natural map. Because of this, homological algebra and derived functors become useful in surprising ways. This thesis is an expository work that provides an overview of the theory of Grothendieck groups with respect to these decategorications. 2015-10-08T19:56:18Z 2015-10-08T19:56:18Z 2015 2015 Thesis http://hdl.handle.net/10393/33000 http://dx.doi.org/10.20381/ruor-2842 en Université d'Ottawa / University of Ottawa |
collection |
NDLTD |
language |
en |
sources |
NDLTD |
topic |
additive categories abelian categories exact categories triangulated categories Grothendieck groups category theory cochain complexes homotopy category cohomology bounded derived category derived functors |
spellingShingle |
additive categories abelian categories exact categories triangulated categories Grothendieck groups category theory cochain complexes homotopy category cohomology bounded derived category derived functors McBride, Aaron Grothendieck Group Decategorifications and Derived Abelian Categories |
description |
The Grothendieck group is an interesting invariant of an exact category. It induces a decategorication from the category of essentially small exact categories (whose morphisms are exact functors) to the category of abelian groups. Similarly, the triangulated Grothendieck group induces a decategorication from the category of essentially small triangulated categories (whose morphisms are triangulated functors) to the category of abelian groups. In the case of an essentially small abelian category, its Grothendieck group and the triangulated Grothendieck group of its bounded derived category are isomorphic as groups via a natural map. Because of this, homological algebra and derived functors become useful in surprising ways. This thesis is an expository work that provides an overview of the theory of Grothendieck groups with respect to these decategorications. |
author2 |
Savage, Alistair |
author_facet |
Savage, Alistair McBride, Aaron |
author |
McBride, Aaron |
author_sort |
McBride, Aaron |
title |
Grothendieck Group Decategorifications and Derived Abelian Categories |
title_short |
Grothendieck Group Decategorifications and Derived Abelian Categories |
title_full |
Grothendieck Group Decategorifications and Derived Abelian Categories |
title_fullStr |
Grothendieck Group Decategorifications and Derived Abelian Categories |
title_full_unstemmed |
Grothendieck Group Decategorifications and Derived Abelian Categories |
title_sort |
grothendieck group decategorifications and derived abelian categories |
publisher |
Université d'Ottawa / University of Ottawa |
publishDate |
2015 |
url |
http://hdl.handle.net/10393/33000 http://dx.doi.org/10.20381/ruor-2842 |
work_keys_str_mv |
AT mcbrideaaron grothendieckgroupdecategorificationsandderivedabeliancategories |
_version_ |
1718598402095185920 |